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The connection between mathematics and art goes back thousands of years. Mathematics has been used in the design of Gothic cathedrals, Rose windows, oriental rugs, mosaics and tilings. Geometric forms were fundamental to the cubists and many abstract expressionists, and award-winning sculptors have used topology as the basis for their pieces. Dutch artist M.C. Escher represented infinity, Möbius bands, tessellations, deformations, reflections, Platonic solids, spirals, symmetry, and the hyperbolic plane in his works.

Mathematicians and artists continue to create stunning works in all media and to explore the visualization of mathematics--origami, computer-generated landscapes, tesselations, fractals, anamorphic art, and more.

The intersections between origami, mathematics, and science occur at many levels and include many fields of the latter. We can group these intersections into roughly three categories: Origami mathematics, which includes the mathematics that describes the underlying laws of origami; Computational origami, which comprises algorithms and theory devoted to the solution of origami problems by mathematical means; Origami technology, which is the application of origami (and folding in general) to the solution of problems arising in engineering, industrial design, and technology in general. One genre blends into another. Origami math defines the "ground rules" for computational origami's goal of solving origami design problems (and quantifying their difficulty). The results of computational origami, in turn, can be (and have been) pressed into service to solve technological problems ranging from consumer products to the space program. Origami, like music, also permits both composition and performance as expressions of the art. Over the past 40 years, I have developed nearly 600 original origami compositions. About a quarter of these have been published with folding instructions, which, in origami, serve the same purpose that a musical score does: it provides a guide to the performer (in origami, the folder) while allowing the performer to express his or her own personality. My website includes galleries of my designs, crease patterns, schedule of my lectures, appearances and exhibitions, commissioned works, and more on the science of origami.

The 2018 Mathematical Art Exhibition was held at the Joint Mathematical Meetings held in San Diego, CA. Here on Mathematical Imagery is a selection of the works in various media, including recipients of the 2018 Mathematical Art Exhibition Awards: "A Gooseberry/Fibonacci Spiral,” by Frank A Farris, awarded Best photograph, painting, or print; "Dodecahedral 11-Hole Torus," by David Honda, awarded Best textile, sculpture, or other medium; and "Excentrica" by Ekaterina Lukasheva, Honorable Mention. The Award "for aesthetically pleasing works that combine mathematics and art" was established in 2008 through an endowment provided to the American Mathematical Society by an anonymous donor who wishes to acknowledge those whose works demonstrate the beauty and elegance of mathematics expressed in a visual art form. Click on the thumbnail images to view larger.

I had always known the beauty and power of mathematics, but after discovering the connection between complex numbers and fractals, I became deeply fascinated in how fractals express and reflect that beauty. Whereas mathematical beauty often lay in abstract concepts and elegant proofs, fractals bring the beauty of mathematics plainly to the surface, unobscured and for all to see. Following my curiosity, I wrote a Java program to generate and explore fractals, as I find that creation and experimentation often brings about a deeper sense of understanding than just observation. At the time I had just tackled a number theory problem regarding tetration (super-exponential functions), and decided to generate fractals based on that. After generating the fractal, I experimented with changing my rendering method a little bit, and to my surprise, a completely new form of fractals emerged, which is shared here. The base function for all these images is still simply standard tetration, the only difference being that color (or shade) of each pixel (representing a point in the complex plane) is dependent on the maximum reference angle of that point as tetration is repeatedly applied, rather than whether or not the magnitude of that point would increase without bounds, the latter being what standard tetration fractals are made from. I was astonished that such a small change would produce such intricate results that were quite different from normal tetration fractals. No matter how much I think about it, I am still amazed that the basis of these images is purely mathematics, and I hope to share that beauty here in this gallery. --- Stephen Ren, North Hollywood, CA

The 2017 Mathematical Art Exhibition was held at the Joint Mathematical Meetings held in Atlanta, GA. Here on Mathematical Imagery is a selection of the works in various media, including recipients of the 2017 Mathematical Art Exhibition Awards: Fractal Monarchs," by Doug Dunham and John Shier, was awarded Best photograph, painting, or print; "Torus," by Jiangmei Wu, was awarded Best textile, sculpture, or other medium; and "AAABBB, two juxtapositions: Dots & Blossoms, Windmills & Pinwheels," by Mary Klotz, received Honorable Mention. The Award "for aesthetically pleasing works that combine mathematics and art" was established in 2008 through an endowment provided to the American Mathematical Society by an anonymous donor who wishes to acknowledge those whose works demonstrate the beauty and elegance of mathematics expressed in a visual art form. The thumbnail images in the album are presented in alphabetical order by artist last name.

I'm particularly interested in visualizing mathematics and giving talks on mathematics and art. Many of my digital works, some of which are made into fabric and wallpaper, are based on photographs of everyday scenes and objects. --- Frank A. Farris, Santa Clara University

After receiving a Ph.D. in mathematics at Ohio State, I taught for several years at Merrimack College in Massachusetts, followed by a visiting position at Hamilton College, before settling into teaching mathematics and computer science at Hopkins School in New Haven, Connecticut. Although I teach mathematics for a living, I am also passionate about coding, music, and visual arts. I have helped to maintain the Flash tutorial site flashandmath, and more recently my own HTML5 Canvas and JavaScript blog, rectangleworld. --- Daniel Gries (Hopkins School, New Haven, CT)

I create geometric patterns in the snow, walking along the frozen lakes of Savoie, France in snowshoes. On average the works take about 10 hours to really do it properly, some are a little unfinished, if my feet get cold or hurt too much. The setting out is done using handheld orienteering compass and distance determination using pace counting or measuring tape. Curves are either judged or arcs of circles are made using a clothesline attached to an anchor at the centre. Designs are chosen from the world of geometry. The Koch curve and Sierpinski triangle in this album are among my favorites. The works are very large (the size of several soccer fields), and many of the mathematical patterns appear 3D, especially when viewed from above. more recently I've also created patterns in the sand. --- Simon Beck

Since high school I have been fascinated by geometry. I enjoyed constructing the more complicated Platonic solids with ruler and compasses, as well as reading about the 4th dimension. While at Bell Labs in Murray Hill, I was introduced to the field of Computer Graphics, and later developed the Berkeley UniGrafix rendering system, so that I could depict objects more complex than I could build. Since then, the focus of my work has been on computer-aided design (CAD) tools -- for engineers, architects, and artists. When creating abstract sculptures I see myself as a composer in the realm of pure geometry. The artistic achievement then lies in finding a procedural formulation that can reflect the inherent symmetries and constructive elegance that seems to lie beneath many sculptural master pieces as well as at the foundations of the physical laws of our universe.

One of my goals is to create very beautiful images by using mathematical concepts such as trigonometric functions, exponential function, regular polygons, line segments, etc. I create images by running my program on a Linux operating system. --- Hamid Naderi Yeganeh

The 2016 Mathematical Art Exhibition was held at the Joint Mathematical Meetings held in Seattle, WA. Here on Mathematical Imagery is a selection of the works in various media, including recipients of the 2016 Mathematical Art Exhibition Awards: "45 Poppies," by Karl Kattchee was awarded Best photograph, painting, or print; "Sword Dancing," by George Hart was awarded Best textile, sculpture, or other medium; and "OSU Triptych No. 2," by Robert Orndorff received Honorable Mention. The Award "for aesthetically pleasing works that combine mathematics and art" was established in 2008 through an endowment provided to the American Mathematical Society by an anonymous donor who wishes to acknowledge those whose works demonstrate the beauty and elegance of mathematics expressed in a visual art form. The thumbnail images in the album are presented in alphabetical order by artist last name.

Weavers of beads use a needle and thread to sew beads together to make decorative objects including jewelry, wall hangings, sculptures, and baskets. Some bead weave designers weave beads into composite clusters, usually with at least one large hole, called beaded beads. Mathematically, many beaded beads can be viewed as polyhedra, with each bead (or, more precisely, the hole through the middle of each bead, which provides its orientation) corresponding to an edge of the polyhedron. Different weaving patterns will bring different numbers of these "edges" together to form the vertices of the polyhedron. So it is very natural to use various polyhedra as the inspiration for beaded bead designs. Mathematics, including geometry, symmetry, and topology, is an inspiration for the structure of these woven bead creations. Across cultures and continents, humans show a natural affinity towards the aesthetic of pattern and order, and this art form appeals to this aesthetic in a tactile, tangible form. --- Gwen L. Fisher (www.beadinfinitum.com)

My work is composed primarily of computer generated, mathematically-inspired, abstract images. I draw from the areas of geometry, fractals and numerical analysis, and combine them with image processing technology. The resulting images powerfully reflect the beauty of mathematics that is often obscured by dry formulae and analyses. An overriding theme that encompasses all of my work is the wondrous beauty and complexity that flows from a few, relatively simple, rules. Inherent in this process are feedback and connectivity; these are the elements that generate the patterns. They also demonstrate to me that mathematics is, in many cases, a metaphor for the beauty and complexity in life. This is what I try to capture. --- Kerry Mitchell

Mathematicians David Griffeath (University of Wisconsin-Madison) and Janko Gravner (University of California, Davis) have built a model that generates detailed 3D images of all types of nature's snowflakes.

Inspired by William Thurston's paper creations back in the 1960s, I thought if something can be made out of paper, it can also be crocheted, so I made my first crocheted hyperbolic planes in June 1997 by increasing stitches in constant ratio---after every two stitches I did an increase by one stitch. The number of stitches in each row grew exponentially, so after finishing my first small, very ruffled one I realized that to explore the hyperbolic plane I have to change the ratio of increase. For classroom use the best is to use the ratio 12:13---it means to increase one stitch after every 12 single crochet stitches. See more crochet examples on my blog, Daina Taimina Fiber Sculptures, at http://dainataimina.blogspot.com/. --- Daina Taimina (Cornell University, Ithaca, NY)

The 2015 Mathematical Art Exhibition was held at the Joint Mathematical Meetings held in San Antonio, TX. Here on Mathematical Imagery is a selection of the works in various media, including recipients of the 2015 Mathematical Art Exhibition Awards: "Penrose Pursuit 2," by Kerry Mitchell was awarded Best photograph, painting, or print; "Map Coloring Jewelry Set," by Susan Goldstine was awarded Best textile, sculpture, or other medium; and "15 Irregular Hexahedra," by Aaron Pfitzenmaier received Honorable Mention. The Award "for aesthetically pleasing works that combine mathematics and art" was established in 2008 through an endowment provided to the American Mathematical Society by an anonymous donor who wishes to acknowledge those whose works demonstrate the beauty and elegance of mathematics expressed in a visual art form. The thumbnail images in the album are presented in alphabetical order by artist last name.

The National Science Foundation and Popular Science are cosponsors of the long-running Visualization Challenge, now called The Vizzies, to recognize some of the most beautiful visualizations from the worlds of science and engineering. "Some of science's most powerful statements are not made in words. From DaVinci's Vitruvian Man to Rosalind Franklin's X-rays, science visualization has a long and literally illustrious history. To illustrate is to enlighten! Illustrations provide the most immediate and influential connection between scientists and other citizens, and the best hope for nurturing popular interest. They are a necessity for public understanding of research developments." --- National Science Foundation

I'm a math teacher, illustrator, and dad. Having begun entertaining my children with weekly pancakes earlier this year, I'm always looking for new themes; in this album you'll find some of the fractal pancakes I cooked up one morning. On my blog, www.10minutemath.com, you can find more about fractals and other topics that interest me as a teacher. --- Nathan Shields (www.10minutemath.com)

The 2014 Mathematical Art Exhibition was held at the Joint Mathematical Meetings held in Baltimore, MD. Here on Mathematical Imagery is a selection of the works in various media. Mathematical Art Exhibition Awards were given: "Enigmatic Plan of Inclusion I & II," by Conan Chadbourne was awarded Best photograph, painting, or print; "Three-Fold Development," by Robert Fathauer was awarded Best textile, sculpture, or other medium; and "Blue Torus," by Faye E. Goldman received Honorable Mention. The Award "for aesthetically pleasing works that combine mathematics and art" was established in 2008 through an endowment provided to the American Mathematical Society by an anonymous donor who wishes to acknowledge those whose works demonstrate the beauty and elegance of mathematics expressed in a visual art form. The thumbnail images in the album are presented in alphabetical order by artist last name.

The 2013 Mathematical Art Exhibition was held at the Joint Mathematical Meetings held in San Diego, CA. Here on Mathematical Imagery is a selection of the works in various media. Mathematical Art Exhibition Awards were given: "Bended Circle Limit III," by Vladimir Bulatov was awarded Best photograph, painting, or print; "Inlaid Wooden Boxes of Makoto Nakamura's Tessellations," by Kevin Lee was awarded Best textile, sculpture, or other medium; and "Tessellation Evolution," a beaded necklace by Susan Goldstine received Honorable Mention. The Award "for aesthetically pleasing works that combine mathematics and art" was established in 2008 through an endowment provided to the American Mathematical Society by an anonymous donor who wishes to acknowledge those whose works demonstrate the beauty and elegance of mathematics expressed in a visual art form. The thumbnail images in the album are presented in alphabetical order by artist last name.